Digital geometry

Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space.

Simply put, digitizing is replacing an object by a discrete set of its points. The images we see on the TV screen, the raster display of a computer, or in newspapers are in fact digital images.

Its main application areas are computer graphics and image analysis.

Main aspects of study are:

Digital geometry heavily overlaps with discrete geometry and may be considered as a part thereof.

A 2D digital space usually means a 2D grid space that only contains integer points in 2D Euclidean space. A 2D image is a function on a 2D digital space (See image processing).

In Rosenfeld and Kak's book, digital connectivity are defined as the relationship among elements in digital space. For example, 4-connectivity and 8-connectivity in 2D. Also see pixel connectivity. A digital space and its (digital-)connectivity determine a digital topology.

In digital space, the digitally continuous function (A. Rosenfeld, 1986) and the gradually varied function (L. Chen, 1989) were proposed, independently.

A digitally continuous function means a function in which the value (an integer) at a digital point is the same or off by at most 1 from its neighbors. In other words, if x and y are two adjacent points in a digital space, |f(x) − f(y)| ≤ 1.

A gradually varied function is a function from a digital space ${\displaystyle \Sigma }$ to ${\displaystyle \{A_{1},\dots ,A_{m}\}}$ where ${\displaystyle A_{1}<\cdots <A_{m}}$ and ${\displaystyle A_{i}}$ are real numbers. This function possesses the following property: If x and y are two adjacent points in ${\displaystyle \Sigma }$, assume ${\displaystyle f(x)=A_{i}}$, then ${\displaystyle f(y)=A_{i}}$, ${\displaystyle f(x)=A_{i+1}}$, or ${\displaystyle A_{i-1}}$. So we can see that the gradually varied function is defined to be more general than the digitally continuous function.

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